The human skeleton is made up of a torso, limbs, and articulation points, each of which can move independently of each other to some extent and also together. In animating a human skeleton, inverse kinematics, or sometimes referred to as "goal-directed motion", is one method of approaching animating articulated chains. Forward kinematics is the other method of animating articulated chains.
In forward kinematics, the skeleton is moved into position by specifying the angle of each joint and then keyframing these positions. Each movement is carefully planned to create the desired animation. Using forward kinematics, the orientation of each joint is precisely controlled allowing creation of many types of movements that may not be possible with inverse kinematics alone.
In inverse kinematics, the goal of an action is defined by positioning an end effector of an articulated chain. An inverse kinematics algorithm then calculates the position of the rest of the chain to reach the desired goal. For example, an animator may wish to animate a character reaching for a glass; the animator positions the wrist or end effector of an articulated chain representing the arm at a location in space where the glass is positioned. The inverse kinematics algorithm calculates the positions and orientations of the rest of the arm as it moves so that the final position has the wrist at the glass.
Inverse kinematics solves for the position and the orientation of all articulation points in the limb hierarchy that lead to the end effector. EQU .theta.=.function..sup.-1 (X)
where .theta. is a state vector and X is a position and orientation in space of the end effector. The dimension of the state vector .theta. is equal to the degrees of freedom of the articulated chain.
The function .function.( ) is highly non-linear and rapidly becomes more complex as the number of limbs increase. In order to invert this function, the function is made linear by localizing about a current operating position and inverting the Jacobian to give: EQU d.theta.=J.sup.-1 (dX)
where J is a matrix of partial derivatives relating differential changes of .theta. to differential changes in X. The function is iterated towards the goal over a series of incremental steps.
Inverse kinematics theory is more fully described in Advanced Animation and Rendering Techniques, Theory and Practice, A. Watt and M. Watt, Addison-Wesley, 1992, New York, Chapter 16, Animated Articulated Structures, pp 369-384, the contents of which are hereby incorporated by reference.
Inverse kinematics is relatively more computationally expensive than forward kinematics. However in animating walking and arm movement, forward kinematics is counterintuitive and tedious to do in practice. Inverse kinematics is thus preferred in many animation applications.
In computing a needed position and/or an orientation of each limb or link of an articulated chain so that the effector reaches the given goal, there can be computed a solution comprising a set of all positions and/or orientations of all limbs. In most cases, there exists many more than one solution for a given goal. For example, for an arm of a character to reach for a glass, the arm could either move across the chest or over the head. The inverse kinematics algorithm must then choose one of the possible solutions. Unfortunately, the chosen solution may not be the solution desired by the animator. Even worse, the inverse kinematics algorithm may choose a first solution for part of the animation and then "flip" to another solution for another part of the animation. The result is that the animated motion will be discontinuous, which is undesirable.
One method of addressing the discontinuity problem is to use a previous position or frame for calculating the next position of the articulated chain. The inverse kinematics algorithm is then limited in the number of possible solutions for the next position thereby avoiding discontinuities. The drawback of this approach is that animation is not fully reproducible. If any errors in the calculations had been made, such errors accumulate. The result becomes apparent when reciprocally extending an articulated chain; the articulated chain is unlikely to return to its original position.